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A linear representation of the mapping class group M and the theory of winding numbers
Topology and its Applications 43 (1992) 47-64 North-Holland
47
A linear representationof the mapping class group A and the theory of winding numbers Rolland ‘I’rapp Dqmrtment of Mathematics, UtWrsit,
of Michigann,Ann Ati,
Ml, 4dW9-1m3, USA
Received 25 Septtmber 1990
Abstmct Trapp, R, A linear representation of the mapping class group numbers, Topology and its Applications 43 (1992) 47-64.
andthe
d-W
This paper describes @ of the mapping surfaceSwithonebo representation @ e tion, and is de&d for surfaces of azbitrary genus g> L The crossed homomorphisms 4%:AVw H’(S; 2) which are de&ted usi X on 8 and the theory of winding numbers of curves on surfaces [I, 21. These cmssd homomorphisms were essenti&y described Q, is then given. If T,S denotes the unit tangent on H,(T,S, 2). The kemel of @ is then characterized using knowledge of the crossed homomorphisms 4%.If matrix entries are taken modulo Zg - 2, the represcnta tion Q factors through the mapping class group 3 of a closed orientable surf&cc of genus g > 1. Thus #Pinduces representations of & of 2 for any n [2g -2. The &m were discovered by Sipe in [7,8], and it is noted that her characterization of the image of @m carries over to the integer valued case. The stnlaure found in characterizing ket @ is then used to study ker &. In particular, it is shown that a quotient of ker 6” is a semidirect product for each even n div@yg 2g -2.
Keywonlu; Crossed homomorphism, winding numbers, mapping class group. AMS (MOS) Subj. Cks.:
Primary 57M99, secondary 55M25,57NO7.
Introductioll
This paper describes a linear representation @ of the mapping class group an orientable surface S with one boundary component. Let HI = H,(S; Z). Then acts on HI and this action is given by the well-known sympfectic representation P:
Jt *
0166-8641/92/$05.00
Sp(2g;
a,
Q 1992-Elsevier Science Publishers B.V. Gil tights resq~ed
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48
where g is the genus of SI The Torelli group, .%,is the subgroupof d acting trivially on ~~ ; in other words, .% = ker p. The representation @ extends p, and is defined forsurfacesof genus g > 1. The main tools used to define @are crossedhomomorphisms e,: Jccw H ‘(S; Z) which are defined using nonvanishing vector fields X on s, and winding numbers of curves on surfaces described by Chillingworth in [1,2]. &~en a nonvanishing, smooth vector field X on S, and a curve y on S, the winding number of 7 with respect to X can be intuitively defined as the number of times
t vector to y rotates with respect to the X-vector as y is traversedonce The notion of winding numbers is used, in turn, to define crossed - H ‘(S; Z). Intuitively,forfE 4f, the class e,(f) E H’(S; 2) thewinding numbers of homology classes with respectto X The fact that the e, are crossed homomorphismsis preciselywhat allows for the t should be noted that, even though e,(f) definition of representations@jXof winding numbers of homologous curves is well defined on homology classes, ly equal. Hence the action of on winding numbers is not trivial. are not The use of vectorfieldsin the definitionof aJpsuggestsa geometricinterpretation. bundle of & and fi, = H,( T,S; 2). Given any f e 4, its Let T,Sbetheunit derivative watts and one would expect a representationof JYin terms of this action. The difficultyin finding such a representation arises in choosing a basis for &, and then calculatingthe action of Qf on this basis. The vector field X will be used to lift curves on S to curves in T’$‘,and the crossed homomorphism e, faci the nec;essarycalculations. Theorem 2.2 then shows that, for any f e A& es thy action of Df on fi, with respect to a particularbasis. A different Q)x( choice of vector field correspondsto a change of basis for I&. Thus the representations & are all conjugate,and are considered a single representation@. Since Qiextends the symplectic representation p, it is immediate that ker @t 9. Moreover,the fact that the action of .% on winding numbers is nontrivial implies that ker @ is a proper subgroup of 9. In order to understand ker @, then, it is nece=y
to study @I,. The representation @ is calculated on an infinite set of
generators for 9, giving rise to a characterizationof ker @. Let S be the closed surface obtained from S by attaching a Z-disc aiang 5as boundary,and 2 its mapping class group. Let p’: 2 I+ Sp(2g; 2) denote the corresponding symplectic representation, and .@= ker p The groups & and 2 are closely related. In particular,if 7$ is the unit tangent bundle of s, then Johnson showed in [S] that the followingsequence is exact: l+rr,(TI~)+.&&#+l.
(1.1)
Moreover, the subgroup ?rl(T,s) of .& is actually a subgroup of 9, giving the exact sequence l+q(T,S)+4+&1.
(1.2)
A natural question arises. Does the representation @ factor through .a? The calculations of @I9 show that Cpfactors through d when the matrices @(f) are
The mapping class group At
49
taken with entries in Z, ratherthan Z for any n I2g - 2 (where g is the genus of S). Let Qzndenote the representation Qi with matrix entries taken in 2,. For n I2g -2, then, the representations @” induce representations @” of a. Sipe discovered the representations @,,in [7], and characterizedthe ihnageof 6” in [8]. It is noted that Sipe’s characterization of im 6” carries over to im QT,Following the notation in [7,8], let Ng.”be the congruencesubgroupof level n in Sp(2g; Z), and Gsn = ker &. Sipe shows that jj( Can) = Nan. The characterizationof ker 4 is used to describe g&n= G&,n 9, obtaining the exact sequence W&,+G~,-+N~,-,f,
(13)
Moreover,it is shown that, for n even and dividing 2g - 2, a quotientof G,, = ker & is a semidirect product. It is interesting to note that in [gf Sipe shows that im & is a semidirect product for n odd, and that a quotient of ker & is a semidirect product for n even. T’he paper is organixed as follows. Section 1 is preliminaryin nature,describing the tools needed to define @.Section 2 provides the main treatmentof the tion @,and Theorem2.2 gives the desired geometric interpretationof ax. of the crossed homomorphisms e, is used to calculate @ on generatorsof 3; obtaining a characterization of ker @. The calculations are then furtheredto more general diffeomorphismsin §. The relationship between the 4i&of J@discovered by Sipe and the representation @ of .4Xis studied in Section 3. In particular, the results of Section 2 are used to reveal more of the structureof the groups Gsn = ker @“. Remark. The representation Cpwas actually found in a somewhat di&rent context from that described in this paper. Squier, in [9], describes a method of defining two-parameterrepresentationsof Artin groups. The group A admits a presentation with “Artin group” and two extra relations (see [lO])- The question of when these two extra relationsare satisfiedwas asked. This happensprecisely whenthe specialization t = -s-* is made, where (0,s) are the parameters itg $ juier’s work. In that case, it turned out that the remaining parameter occured only trivially, and the resulting representation was Qz.From this point of view, Theorem 2.2 is surprising indeed.
1. Preliminaries and notation In this section the necessary background information and definitions are given. Mapping class groups are discussed m Section Ll, with an emphasis on the Torelli group. Section 1.2 deals with the unit tangent bundle of a surface with one boundary component and conventions about vector fields. Winding numbers are the topic of Section 1.3. Here the crossed homomorphisms e, are defined, and some of tGr properties discussed. This material, although it may be the least familiar POthe
50
R TMPP
reader,is the most importantin what is to come. Here is a summary of the notation introducedso far: o s a closed orientablesurface of genus g> 1, l S = $02-disc, 0 J(d,3 the respectivemapping class groups, . n* = H,(S; 2) = H,(S; 2); H’ = H’(S; Z) = If’@; e ~5@the respectivesymplectic repmentations, with kernels4,& ve unit tangent bundles, smooth, nonvanishing vector field on 8
however, this referencemay be helpful. 1.1.
Given s S as above, the CurveS{ati,6 6}i=a,_d, pictured ia Rg. l(a) will be called generatorsfor both q((s, *) and q(& *). A collectionof simple closed basis for HI if (Ui,aj)=O=(bi, bj), cUWs iS, bili==*,...dtill be called a SYII@& and (pi, b)= a@9 where ( ,) denotes the intersectionpairing in &. The projections (e, &+}of the CUW~S {ai, &} form a symplectic basis.
7he mapping class group A
51
Recall that the representation @, which will be described, extends the symplectic representation; therefore, the kernel of @ is contained in .%.In order to understand ker @, it is necessary to know more about the structure of the group .%.Toward this end, two special types of curves on S are distinguished which will give rise to important classes of maps in §. DelBaItIon1.1.1. A simple closed curve y on S is a hn&g simple closed ame (BSCC) if it bounds a subs&ace Sp S. The genus of S,, is the genus g(y) of y (see Fig. l(b)). De&&Ion 1.1.2. A pair of disjoint, homologous, simple cl& curves ( y, S) is called a bn&g puir (BP) if y is not homologous to 0. Then the pair (y, 8) bound a subsurface Sy,a,and the genus of the bounding pair is the genus of &a (see By conventions, the subs&ace ively Sr.&) is chosen which y (respectively (‘y,8)) boun oes not contain the curves just defined give rise to the following diffeomorphisms in .
Dedidon
1.1.3.A Dehn twist about a BSCC y is called a BSCC
denoted T7. Defdtion 1.1.4.A bounding pair map (BP map) is collfip;hsedof
T, 7’;’ about a BP (y, 6). The genus of a BP map is the genw of the Johnson showed in [3] that, for g 23, 4r is actually generated by BP maps of genus one. To derive the corresponding result for 9, ga 3, one notes that 3 is a quotient of .% (see exact sequences (l-l), (1.2)). Thus 3 is also generated by BP maps of genus one, for g a 3. Moreover, in the process of proving the exactness of sequence (1.2), Johnson shows that n,( T$) c 9 i ge~rated by Wrnqs of maximal genus; i.e., of genus g - 1. This fact will be used in determining how to obtain =. representations of ##frarn G! The case g = 2 is a bit different. In this case, Johnson showed that 9 is generated by BP maps of genus one together with BSCC maps. Note that when g = 2, any BP map is of maximal genus, thus P,( T,s) is the group of BP maps. The subgroup of .% (respectively 3 ) generated by all BSCC maps will be denoted by X (respectively 9). It will be shown that Xc ker @; therefore, BP maps of genus one will be of primary importance when calculating @I4 for all g 2 2. 1.2.
Some properties of the unit tangent bundle, which will be needed later, are reviemd in this paragraph. Let S have some Riemannian structure, inducing a IIO !! iju
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52
on each fiber IsP of the tangent bundle X9. Then T,S is the collection of all tangent vectors of unit length, i.e., T,S=
u
(UE =p/ Il4lp= 1).
P-
The unit tangent bundle T,S of S is defined similarly. It is well known that T,S is the trivial bundle, or T,S= Sx S’; hence, G, = pl( T&5)= q(S) x and fi,= At times it will be useful to think of H, asa subgroup of & in this
manner, In terms of exact squen
9 that above
14+&H,-4,
description gives (1.2.1)
where the kernel is generated by the fiber dass [z] E fi,. In the case of the closed of T$‘, however, is easily rds, the fiber class [z] E #ii vial bundle, S admits nonvanishing, mooth vector field on s it induces a on of T,S by the formula
the vector field and the section will be denoted by X) In what is to follow, X on S 41 mean a global smooth section of T,S A formula similar be used to define the diffeomorphism Qf of T,S induced by f~ 4 The derivativewoffinduces a difkomorphism of T,&,alsodenoted by Df, defined
by (l-2.3) Via dierentiation, then, an action of A on fil is obtained, and this action will be calculated. One anticipatory remark is in order before discussing winding numbers. Since & = %+‘, the action of Dfon a1 will be given by a (2g + 1) x (2g + 1) matrix with integer coefficients. Moreover, Df wiil always preserve the fiber class [ t’j E a,. If &, 6 are lifts of ui, bi to @, then the matrix of w with respect to the basis {a, a’,, . . ..&.d ,,..., gg} of fi, will have the vector (l,O, . . . ,0) as its first column. 1.3. In this section, the definition of winding numbers of curves on surfaces is given, as well as a review of some of their properties. An interpretation of winding numbers in terms of intersections in T,S is also given. This definition is particularly useful for the calculations required in the proof of Theorem 2.2. The crossed homomorphism e, is then defined, and some of its properties outlined. In particular, the behavior of the e, under composition in J# is calculated. This formula will suggest a way to define representations QXof A. First, Remark 2 on p. 221 of [l] is reproduced.
7&e mapping class group A
53
Let y be a regular closed curve on S which is parameterized by arc length (i.e., Ily’(N,W= 1 for all t ). Then T,SI, is a torus T*. Let X be a vector field on S and X(y) * Xl,,, then it is easy to see that X(y) is a curve on T*. Choose an orientation on T,S. Let a be the homotopy class of X( y) and @the homotopy class corresponding to a fiber with induced orientation; then, nl( T*) = (a, /3 1tujh~“‘/3’~= 1). Since y is parameterized by arc length, the tune 7 = ( y(t), y ‘( t )) lies on T*,and p = jslllQtE rrl( T*) for some integer m, Chiilingworth defines &y) = RP& Note that the sign of 111depends on a choice of orientation for T,S. Hodgson suggested an alternate description of this definition in terms of intersections in the homology of T,S. If X is a smooth global section of T&t, then X(S) is a surface in T,S, i.e., X(S) E H2( T,S, aT,S; ). By pojnc& duality, X(S) E if’, and the value of X(S) on any class [+]e fit is given by (X(S), j$ Choose an orientation for T,S so that (X(S), z)= 1, where [z] is the fiber class with indu orientation. Calculating X(S) on the generators a, j3 of q(r*) gives
wm
(1.34
P?= W(S), g)= 1,
(1.3.2)
(X(S),a)=(X(S),X(y))=O.
Hence (X(S), jQ = (X(S), /3*a) = m, and the following de&titian can be
o,(Y)=wm 3% with the one given This definition, with careful choices of orientations, coin by Chillingworth. This definition of wx(y) is only for curves on the sutiace S;, however, ChiWgworth shows that the notion of windin umber can be extended to the case of the closed surface if values are taken in 2g_t rather than I!. This fact will be useful in determining how the representations ex of JNcan induce representations of 2. Two properties of winding numbers are used in defining the crossed homomorphisms e, : A I--)H’. First, given two vector fields X2 ) X2 on cSi z&e is a well-defined element elJE 1y’ given by (1.3.3)
e,Jrl = o,,(Y) -w,,(Y)
for any [y] E HI. This is well defined on H, c fiI since e1,2 vanishes on the fiber class. Secondly, note that oDJ.,(fy) = w,(y). Definition 1.3.1. Let X be a vector field on S. Then e,
: A k-, H’
is defined by
e,(f)Crl=o,(fy)-o,(y) for any fdt,
ye&.
Here, and for the remainder of the paper? fv denotes &.r where _& L :ke from i automorphism of W,(S) induced by f: The fact that e,(f) E ’ fbfh6
54
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that I, = ~~-a~(y) and the previous discussion. The most important property of the e, is the composition law:
~AfWCrl=o,WwhfhW =w,(fhr)-w,(hy)+o,(hy)-w,(y) (1.3.4)
= ex(fMrl+ ex(wCYl~ Or, more simply stated: e&tWlrl = 4f )Ihyl+ e,(JOCy19 forallf, he
(1.3.4)
and all [y] E H,. Now consider elements u E H’ as vectors in
u = Wol),
l
. . s u(q&
u(b),
l
l
- 9 W&J1
bs} is a synpkctic basis for HR.Consider the action of 36con H’ in terms UE H’ and the sympkctic representation p. The reader can convince f that the following formula is true: WY1 = [U(Ql)r-
l
l
9 u(b,)l
l
(P(h) ’ CYI)
= (bb,), . . 3 N&)1 P(W) [Yl l
l
l
(1.3.5)
= (u PW . ZYI, l
where all products are usual matrix multiplication and [ y] E HI is considered to be r, Now use (13.5) to rewrite (13.4) as c,(fTc) = df 1 P(h)+s,m
(1.3.6)
l
This formula indicates how to use the crossed homomorphism e, to define a representation @&of Before doing so, however, some remarks about exls are in order. An immediate consequence of (1.3.6) is that the exls are homomorphisms since p(h) is the identity matrix for h E .%.The following property of the homomorphisms eXIJwas proven by Johnson in [4]. His argument is repeated here.
lcralna 1.3.2. 7%ehomotmophism e,l$ is independent of the choice of vectorJeld X. Roof. Let X1, X2 be vector fields on S, and f~ .%.Then
lex,(f )-e,,(f MY1 = o,,(fY)
-o,,(Y)
- bx*(fY)
- @xz(Y)l
=~~,(fY)-~~*(fY)-[o,,(Y)-o,,(Y)l = e*z[frl - e,*2tYl = e,*r Yl - e*,z[rl = 0. The last equality holds since f e 9.
(1.3.7)
0
One reason to consider e& is that ker ax c A It will turn out that changing the choice of vector field X gives rise to conjugate representations. Hence ker Gx is independent of the choice of vector field X. Lemma 1.3.2 is just another way of stating this fact.
‘Chemapping class group 4
55
2. The rqmsentation Qi:A t, KS In this section the representationex is defined using the crossed homomorphism e,, and the dependence of ex on the vector field X is studied. The definition of winding numbers is used to show that ex calculates the action of A on fi,. As a result of this geometric interpretation,the image of A under #x is denoted by qx. The representation @I9 is then studied. In particular,the image of .%under QTis characterizedalong with ker a. Debition 2.1. Given a vector field X, the representation & :A c+ vx is given by
(2.1) where p(f) is the image offed considered to be a vector.
under the symplectic representationand e,(f) is
To see that & is indeed a representation,formula (13.6) is used in the fdlw calculation.
W) The following theorem gives the desire&geometric interpretationof the representation @,..
hf. The calculation shall be performed for a”#,all other cases are completely analogous. Suppose p( f )[a,] = (zf=, ttQ ai + ni br) in H%and consider Qf(ir’& l
l
since Qf(z) = z and Df( 6) = f< for any regular closed curve on 2%It is necessary, then, to computefz in terms of the basis {z, l&,di} for &. Note that, sincefar and (t: FzInti Ei + ni 6 ) project to the same homology class in Hz, l
l
i=l
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56
for some integer n. To calculate n, intersect both sides of equation (2.4) with X(S) E fi’. By the definition of winding numbers, the equality
(
X(S), fx-
f mi i=l
l
i& +
ni 6
=(X(S),
l
nz)
>
becomes
(2.5) Combining (2.4), (2.3, and using the definition of {&, &}, gives
With
this
formula (2.3) becomes
cahhtio~
(2.7)
= f mi4i+ni*&+ex(f)[a,]*Z=@x(f)*[ZJ. i=l
In a similar manner it is shown that Df( 9) = @J f)
l
[f]
for all [ +] E fi, . q
An immediate consequence of Theorem 2.2 is the following
C~rdlUy 23. If Xl v X2 ate vectorfields on S, then the representations dQ,, tBs2 u”,‘c” conj44gate. Roof. By the theorem, both ax, and @x2describe the action of A on & with respect to the appropriate bases; therefore, they are conjugate. El Since the representations eX are all conjugate, they are considered a single representation a. Note, however, that the group S& depends on the choice of X. From Definition 2.1 it is easy to see that the representation @ extends the symplectic representation as promised. In order to describe ker @, it is necessary to consider @I$. For f c JJ, definition (2.1) gives
l3e mapping class group d
Here Id is the 2g x2g identity matrix. Recall that the homomorphism e,k is independent of the choice of vector field; hence, the following results and calculations are independent of the choice of vector field. The first task is to calculatethe matrix 4&(T,T$), where (a~,ai) is the BP of genus one pictured in fig. 2. If fo = T,T$ then fo(q) is conjugate to CQfor i + 2; likewise,fd & ) is conjugateto & for all i Since winding numbers are welt defined on conjugacy CI , it su&es to consider ex(fol[az] in the calculation of e,(&). The formula for calculation of winding numbers given in [I] yields ex(fo)[a,l = -2. Thus, consideringe,&) as a vector gives l%is calculation czn be used to calculate e,( T,T,-‘)
Since such maps generate .%for ga3, the desired ker @ will follow. Propebn2.4.
Let(y,~)benBPofgenwo~
withy
its kjt. Then
e,( T,T,L’) = mG a,),
l
l
l
9
(‘yra,), h b,),
l
l
l
t (Y,b,)).
Proof. For (y, 6) = (ao,a&), equation (2.9) proves the proposition. Now let (y, S) be any BP of genus one, and let f E At be such that fa= 7,fu&= SE. Then T,Ti* = f&,T;;f-‘,cuad
-(; ;g)(;
@xWJi*)-
ex;-fy; ;g,y>
= 1 e,(f-')+e,(f).df-*)+ex(h)=df-*). Id > 0
(2.10)
But O=e,(ff-')=e,(f).p(f-')+e,(f-*), so (2.10)becomes
Once it is shown that e,( fJ p( f -I)= 2(('y, a,), . . . , (y, b,)), the proof will be cornplete. The calculation preceding the proposition showed that l
e,(fo) = W-J,, 4,
- . . 9(a,, b,)) = !e,MJM,
. . 3eLhd?+J~; l
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58
By the choke off, however, fao = -y and (2.11) becomes e,(fo)
l
df-‘1 =2((3r,
QI), . . . 9 h
(2.12)
b,))
as desired. position 2~4facilitates the calculation of e&f) for anyf= T%T&* where (ri, 86) 8te BPS of genus one.
l
Iff = Ta T&’ e,(f) = ex(TI Tc’
l
l
l
TvmTzE
whew each (yip &) is a BP of genus one,
l
l
TvmTz)
(2.13) Prooc, This is a direct consequence of Proposition 2.4 and the fact that exls is a homomorphism. c1 Corollary 2.5 can now be used to characterize ker a. 2.6. The Chillingworth subgroup %‘c § is defined by = f=T,&‘.-
I
Tyn T&l1each ( ‘yiS6i) a BP of genus one, and($,2$=OiuHJ.
Define the subgroups %“nsimilarly, the requirement being that (Cr=, 2x ) = 0 in A)rather than H, . The subgroups %g,nwill be useful in studying the structure of the groups Gg,nof [7,8]. In order to characterize ker @, Corollary 2.5 will be used together with the following fact about the group Xc 9 generated by Dehn twists on BSCCs. Chillingworth shows in [2] that, if y is a SCC, the following formula is satisfied: .
o,(Tys)=ox(B)+(Y, rB) l
o,(Y),
(2.14)
for any curve /3 on S. if y is a BSCC, then (y, ~3)= 0, and formula (2.14) showsthat TyE ker @. Since STis generated by such maps, Xc ker @.
7&emappingclassgroup At
59
Corollary 2.7. For g 2 3, % = ker #. Proof. Recalling that 9 is generated by BP maps of genus one, this is a direct consequence of Corollary2.5 and Definition 2.6. 0
Remark. The case g = 2 is special. For the purposes of this paper, it sufficesto say that it can be shown that X= ker @. Corollary 2.5 can also be used to characterizeim @I$. n2.8. Forga2, if v E (2z)2=.
Pro
thematrix (’O Id) ff is in the image of 4 under @ ifand ody
Proof. Corollary2.5 implies necessity for the subgroup of S gen of genus one. For is implies necessityfor §* Forg = 2, Corolhuy2.S, with the remarks Corollary2.7, imply necessity.Converse1 choices of BP maps of genus one, it is easy to see that contained in im(e,lJ. Since exls is a homomo
In orderto see how @ can induce repesentations of under @ must be studied. Recallthat We genus; hence it is necessary to caiculate @ on Proposition2.9. Let
T,G’
be
a BP map of genus g’. zhen
ex( T,T,‘) = 2g’((y, a,),
l
- 9(Y,a,>, l
where y is oriented so that SYsslies on its lejk Proof. The idea is to rewrite T,Ti’ in terms of BP r the calculation. Let E@, . . . , &8*-t be orientations.Then T’m-1 I+ 8 =(T,T;;)(T,,T;:) * 9 (I&T;;‘), l
of genus one, and use rves pictmed in Fig 3
l
and, using Corollary 2.5, (2.15)
Fig. 3.
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60
However, y is homologous to &i for each i and [ y +C:LI’ Ei] = g’ [ y]. Equation (2.15) becomes l
e,( T,T,-9 = 2g’((y, 4, proving the proposition.
l
l
. 9 (Y, &)I,
(2.16)
0
Note that if (x6) is a BP of maximal genus, then g’= g - 1. Proposition2.9, with the fact that q( T,s) is generatedby BPmaps of maximal genus, imply t Lug 210. For g 3 & the matrix (i &) is in the image of w,( T$) under # ifand 0 E ((2g-2)2)‘9 1 that (p, denote @ with matrix entries taken in 2”. Then Corollary 2.10 the !I&induce representations6” (T$)cker& forn12g-2.
were discovered by Sipe, and are -2. The representations the topic of the next section. is the mapping class group of a once-punctured Onefinal remarkis in order.If sutfaoe, then there is the exact sequence
*+l,
(2.17)
where the kernel is the infinite cyclic subgroup generated by a Dehn twist about of SI Since the representation@ is trivial on BSCC maps, it follows the that oliinduces a representationof “II,,also denoted by @.All the results of this sectioncarryoverto the case of the mappingclass groupof a once-puncturedsurface.
The context in which Sipe discovered the representations& is described in this section, and her characterization of im &,, is mentioned (for details see [S]). This characterization extends to one of ST%, with the appropriate choice of vector field X0 (recall that & depends on the choice of vector field). Yhe results of Section 2 are then applied to gain further insight into the structure of the groups Can. Sipe discovered the representations 6,, while calculating the action of 2 on nth roots of the canonical bundle of the surface g Some of the results found in [7,8], are reviewed here. First, by a Chem class argument, the canonical bundle admits nth roots if and only if n ]2g - 2; therefore, it will be assumed that n I2g - 2, where g is the genus of & Sipe shows that there is a one-to-one correspondence between nth roots of the canonical bundle and the set
This topological description of nth roots is used in what follows. In order to calculate the action of & on nth roots, then, it suffices to know the action of 2 on H,( T,s; Z,). Sipe was able to calculate this action, obtaining the representations &. Recall that
The mapping class group yll
61
@,,denotes the representation @ of A with matrix entries taken in Zn. Furthermore, for n 128-2, the representation @” induces a representation of 2. The induced representation coincides with 8~~since both give the action of J@on ff,( I’,& 2,). Thus Sipe’s results give information about @,,,and it is natural to ask which results carry over to the representation @. Sipe’s characterization of im &, for example, immediately carries over to the integer-valued case. With respect to the basis {z, 4 + z, 6 + t} for fir, the following characterization of the image of 6” was given: w
3.1 (Sipe). 7%e matrix (i i) is contained in the image of & ifad
only if
Diag(B’NB) - ttE (2Z,)‘q
where BE Sp(2g; 2,) and N = (z k’). Here Diag(A) is the 2g vector whose ith component is Uiiand Bt is the transpose of B. Let X0 be the V-O~ field such that Wa(ai)=-t, @a(&)=-I for aEti Tfren, by Theorem 2.2, #% caicuiates the action of on fir with (5 &+S &+Z}* Comolluy 3.2. ne matrix (i i) is in S& ifand only if Diag(B’NB)-1t~(2b)~5 Proof. This is an immediate consequence of Theorem 2-2 and a direct generalization of Sipe’s arguments in [g] to the integer-vaiued case. Cl Note that Proposition 2.8 is the equivalent of Lemma 1 in [S], and that the results OfSection 2 reveal more of the geometric structure of @IsLA&bough Sipe’s arguments regarding im & immediately generalize to the integer-valued case, her discussion of ker @,,= Gsn does not, Let I+&@ c=Sp(2g,P) be the congruence subgroup of level n. Sipe shows that nt&an af $. ‘Ihis result does not p’(G,J = IV&“,where p’ is the symplectic rep apply to the integer-valued case since ker @c A In fact, the zesults of Section 2 can be used to further describe the structure of G”,. The fact that @(G&J = &V&m characterizes the symplectic part of C,,, while the results in Section 2 can be used to characterize G%. n x Recall the exact sequence (1.1) and Definition 2.6 of the groups @&,n. Defieritioa3.3. Let eg,” be the image of %& in .#, and iet @ be the image of the Chillingworth subgroup in J#. An immediate consequence of the definitions is the following Proposition 3.4. Gg.nfl
3 = egvn; hence the following
sequence is exact:
62
R TmPP
Roof. Definition 3.3, Corollary 2.5, and the fact that @” induces 6”.
Cl
The characterization of G”,”fl$ is virtually a direct consequence of properties of @1.,discussed in Section 2. With a little work, however, even more of the structure of Gsn is revealedNote that WM1_-2c G,, foi all R12g-2. In other words, &g-z is the subgroup
of 3 which acts trivially on nth roots of the canonical bundle for all n l2g -2. It wiil be shown tha for n even, the group Gg,,/@3,_2 is a semidirectproduct.The er of the discussion, the% will be restrictedto the case where n is even and rem be to obtain a semidirect product struare on quotients rr12g-2. The strategy n 2, and then show that these using results from of certain subgroups in enotes ker am,then pr(%!&,)= quotientsare isomorphicto G&J @&+ If !Q G,, where pr is as in sequence (1.1). Moreover,as a result of squences (l.l), (1.2), , it follows that p( %!%,) = &,“. Definition2.6 and Spe’s and C&o11 II179, giving rise to the exact squence
for all q and %c 9, the squence (3.2) modulo Ggbecomes
lthat
= ker @; therefore,the groups W&,/W,S&J % can be thought of as
subgroups of the matrix group rp%. Explicit use of Corollary 3.2, the choice of and the fact that n is even, gives the followingdescriptionsof the
(3.4)
qgJ~=@JJ~&“)=
vale,
BEN&, .
(3.3)
Note that since n is even, B E Ng,, implies B = (Id) mod 2, and Diag(B’NB) = Diag((Id)‘N(Id)) = (0,. . . , 0) mod 2.
(3.6)
Hence,if (i L) satisfies conditions (3.9, then it satisfies the conditions of Corollary 3.2. This remark also implies that the matrix (i i) is in @,J $$,) for n even and all B E Ng,*. Thus any matrix (Io B“) E @%( ?$,) can be written Uniquely as a product (i i) (i G), where both (’o B “) , (’o Id“) are in @-+(9&J. In other words, there is a semidirectproduct structure on gg,,/ Ce.Under the obvious identification of %$,/ %’ with (nh)2g the previous discussion gives the isomorophism l
(3.7)
75e mappingclassgroup A
63
Now that the desired semidirect product structure has been obtained on quotients of subgroups of A, it remains to see how this structure projects to subgroups in Recall that the representation @ induces representations of a when matrix entries are taken module 2g -2. Moreover, %sse_2/ %?c %+,,I% for n 128-2. Using the identification of %&.,/% with (nZ)*$ it is clear that
where k = (2g -2)/n. Now consider sequence (3.3) modulo the subgoup Wti~_#4E, obtaining (3.9) which becomes, after the appropriate identifications, 1+
(Zk)*&-+q&J
rg,,_,+NIkn+
(3.10)
1.
Consider %!,/ % to be the semidirect product (nZ)” >Q&+ and note that mo&g out by ((2g -2)2)** only affects the (nZ)*” factor- Hence 4p&n/ ,_2isa product as well. More explicitly,
Thissemidirectp~ductstructureonsubgroupsin~ product structure on subgroups in Propa&ion 3.5. For n emn and n 12g-2,
actually
toasemim
G,,/ @&--z s Cd&)*’>QN&,
(2s.2)/n. Woof. By equation (3.11), all that needs to be shown is that
Since ker(pr) = w,( T,& c %tig-2, the proposition is a consequence of the definitions of g&n and @gG8-2.Explicitly,
Acknowledgement This paper constitutes part of a Ph.D. thesis suf: K&led to Columbia University. The author wishes to thank his advisor Joan Birman for suggesting the problem of specializing Squier’s work, and for the ensuing encouragement and support He is also indebted to Craig Hodgson for the interpretation of winding numbers in %wns of intersection theory.
R. Trtapp
64
Refemms D.R.J.Chillingworth, Winding numbers on surfaces I, Math. Ann. 196 (1972) 218-249. D&J. Chillingworth, Winding numbers on surfaces II, Math. Ann. 199 (1972) 131-153. D. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Sot. 75 (1979) 119-125. n, An Abelian quotient of the mapping class group J$, Math. Ann. 249 (1970) 225-242. n, The structure of the Torelli group I: A finite set of generators for .& Ann. of Math. an manifolds and characteristic classes of surface bundles II, Proc. of the canonical bundle of the universal Teichmiiller cuwe and certain subgroups of a surface, Rot. Amer. Math. Sot. 97
up of an orientable surface, Israel J.
47
A linear representationof the mapping class group A and the theory of winding numbers Rolland ‘I’rapp Dqmrtment of Mathematics, UtWrsit,
of Michigann,Ann Ati,
Ml, 4dW9-1m3, USA
Received 25 Septtmber 1990
Abstmct Trapp, R, A linear representation of the mapping class group numbers, Topology and its Applications 43 (1992) 47-64.
andthe
d-W
This paper describes @ of the mapping surfaceSwithonebo representation @ e tion, and is de&d for surfaces of azbitrary genus g> L The crossed homomorphisms 4%:AVw H’(S; 2) which are de&ted usi X on 8 and the theory of winding numbers of curves on surfaces [I, 21. These cmssd homomorphisms were essenti&y described Q, is then given. If T,S denotes the unit tangent on H,(T,S, 2). The kemel of @ is then characterized using knowledge of the crossed homomorphisms 4%.If matrix entries are taken modulo Zg - 2, the represcnta tion Q factors through the mapping class group 3 of a closed orientable surf&cc of genus g > 1. Thus #Pinduces representations of & of 2 for any n [2g -2. The &m were discovered by Sipe in [7,8], and it is noted that her characterization of the image of @m carries over to the integer valued case. The stnlaure found in characterizing ket @ is then used to study ker &. In particular, it is shown that a quotient of ker 6” is a semidirect product for each even n div@yg 2g -2.
Keywonlu; Crossed homomorphism, winding numbers, mapping class group. AMS (MOS) Subj. Cks.:
Primary 57M99, secondary 55M25,57NO7.
Introductioll
This paper describes a linear representation @ of the mapping class group an orientable surface S with one boundary component. Let HI = H,(S; Z). Then acts on HI and this action is given by the well-known sympfectic representation P:
Jt *
0166-8641/92/$05.00
Sp(2g;
a,
Q 1992-Elsevier Science Publishers B.V. Gil tights resq~ed
R TmPP
48
where g is the genus of SI The Torelli group, .%,is the subgroupof d acting trivially on ~~ ; in other words, .% = ker p. The representation @ extends p, and is defined forsurfacesof genus g > 1. The main tools used to define @are crossedhomomorphisms e,: Jccw H ‘(S; Z) which are defined using nonvanishing vector fields X on s, and winding numbers of curves on surfaces described by Chillingworth in [1,2]. &~en a nonvanishing, smooth vector field X on S, and a curve y on S, the winding number of 7 with respect to X can be intuitively defined as the number of times
t vector to y rotates with respect to the X-vector as y is traversedonce The notion of winding numbers is used, in turn, to define crossed - H ‘(S; Z). Intuitively,forfE 4f, the class e,(f) E H’(S; 2) thewinding numbers of homology classes with respectto X The fact that the e, are crossed homomorphismsis preciselywhat allows for the t should be noted that, even though e,(f) definition of representations@jXof winding numbers of homologous curves is well defined on homology classes, ly equal. Hence the action of on winding numbers is not trivial. are not The use of vectorfieldsin the definitionof aJpsuggestsa geometricinterpretation. bundle of & and fi, = H,( T,S; 2). Given any f e 4, its Let T,Sbetheunit derivative watts and one would expect a representationof JYin terms of this action. The difficultyin finding such a representation arises in choosing a basis for &, and then calculatingthe action of Qf on this basis. The vector field X will be used to lift curves on S to curves in T’$‘,and the crossed homomorphism e, faci the nec;essarycalculations. Theorem 2.2 then shows that, for any f e A& es thy action of Df on fi, with respect to a particularbasis. A different Q)x( choice of vector field correspondsto a change of basis for I&. Thus the representations & are all conjugate,and are considered a single representation@. Since Qiextends the symplectic representation p, it is immediate that ker @t 9. Moreover,the fact that the action of .% on winding numbers is nontrivial implies that ker @ is a proper subgroup of 9. In order to understand ker @, then, it is nece=y
to study @I,. The representation @ is calculated on an infinite set of
generators for 9, giving rise to a characterizationof ker @. Let S be the closed surface obtained from S by attaching a Z-disc aiang 5as boundary,and 2 its mapping class group. Let p’: 2 I+ Sp(2g; 2) denote the corresponding symplectic representation, and .@= ker p The groups & and 2 are closely related. In particular,if 7$ is the unit tangent bundle of s, then Johnson showed in [S] that the followingsequence is exact: l+rr,(TI~)+.&&#+l.
(1.1)
Moreover, the subgroup ?rl(T,s) of .& is actually a subgroup of 9, giving the exact sequence l+q(T,S)+4+&1.
(1.2)
A natural question arises. Does the representation @ factor through .a? The calculations of @I9 show that Cpfactors through d when the matrices @(f) are
The mapping class group At
49
taken with entries in Z, ratherthan Z for any n I2g - 2 (where g is the genus of S). Let Qzndenote the representation Qi with matrix entries taken in 2,. For n I2g -2, then, the representations @” induce representations @” of a. Sipe discovered the representations @,,in [7], and characterizedthe ihnageof 6” in [8]. It is noted that Sipe’s characterization of im 6” carries over to im QT,Following the notation in [7,8], let Ng.”be the congruencesubgroupof level n in Sp(2g; Z), and Gsn = ker &. Sipe shows that jj( Can) = Nan. The characterizationof ker 4 is used to describe g&n= G&,n 9, obtaining the exact sequence W&,+G~,-+N~,-,f,
(13)
Moreover,it is shown that, for n even and dividing 2g - 2, a quotientof G,, = ker & is a semidirect product. It is interesting to note that in [gf Sipe shows that im & is a semidirect product for n odd, and that a quotient of ker & is a semidirect product for n even. T’he paper is organixed as follows. Section 1 is preliminaryin nature,describing the tools needed to define @.Section 2 provides the main treatmentof the tion @,and Theorem2.2 gives the desired geometric interpretationof ax. of the crossed homomorphisms e, is used to calculate @ on generatorsof 3; obtaining a characterization of ker @. The calculations are then furtheredto more general diffeomorphismsin §. The relationship between the 4i&of J@discovered by Sipe and the representation @ of .4Xis studied in Section 3. In particular, the results of Section 2 are used to reveal more of the structureof the groups Gsn = ker @“. Remark. The representation Cpwas actually found in a somewhat di&rent context from that described in this paper. Squier, in [9], describes a method of defining two-parameterrepresentationsof Artin groups. The group A admits a presentation with “Artin group” and two extra relations (see [lO])- The question of when these two extra relationsare satisfiedwas asked. This happensprecisely whenthe specialization t = -s-* is made, where (0,s) are the parameters itg $ juier’s work. In that case, it turned out that the remaining parameter occured only trivially, and the resulting representation was Qz.From this point of view, Theorem 2.2 is surprising indeed.
1. Preliminaries and notation In this section the necessary background information and definitions are given. Mapping class groups are discussed m Section Ll, with an emphasis on the Torelli group. Section 1.2 deals with the unit tangent bundle of a surface with one boundary component and conventions about vector fields. Winding numbers are the topic of Section 1.3. Here the crossed homomorphisms e, are defined, and some of tGr properties discussed. This material, although it may be the least familiar POthe
50
R TMPP
reader,is the most importantin what is to come. Here is a summary of the notation introducedso far: o s a closed orientablesurface of genus g> 1, l S = $02-disc, 0 J(d,3 the respectivemapping class groups, . n* = H,(S; 2) = H,(S; 2); H’ = H’(S; Z) = If’@; e ~5@the respectivesymplectic repmentations, with kernels4,& ve unit tangent bundles, smooth, nonvanishing vector field on 8
however, this referencemay be helpful. 1.1.
Given s S as above, the CurveS{ati,6 6}i=a,_d, pictured ia Rg. l(a) will be called generatorsfor both q((s, *) and q(& *). A collectionof simple closed basis for HI if (Ui,aj)=O=(bi, bj), cUWs iS, bili==*,...dtill be called a SYII@& and (pi, b)= a@9 where ( ,) denotes the intersectionpairing in &. The projections (e, &+}of the CUW~S {ai, &} form a symplectic basis.
7he mapping class group A
51
Recall that the representation @, which will be described, extends the symplectic representation; therefore, the kernel of @ is contained in .%.In order to understand ker @, it is necessary to know more about the structure of the group .%.Toward this end, two special types of curves on S are distinguished which will give rise to important classes of maps in §. DelBaItIon1.1.1. A simple closed curve y on S is a hn&g simple closed ame (BSCC) if it bounds a subs&ace Sp S. The genus of S,, is the genus g(y) of y (see Fig. l(b)). De&&Ion 1.1.2. A pair of disjoint, homologous, simple cl& curves ( y, S) is called a bn&g puir (BP) if y is not homologous to 0. Then the pair (y, 8) bound a subsurface Sy,a,and the genus of the bounding pair is the genus of &a (see By conventions, the subs&ace ively Sr.&) is chosen which y (respectively (‘y,8)) boun oes not contain the curves just defined give rise to the following diffeomorphisms in .
Dedidon
1.1.3.A Dehn twist about a BSCC y is called a BSCC
denoted T7. Defdtion 1.1.4.A bounding pair map (BP map) is collfip;hsedof
T, 7’;’ about a BP (y, 6). The genus of a BP map is the genw of the Johnson showed in [3] that, for g 23, 4r is actually generated by BP maps of genus one. To derive the corresponding result for 9, ga 3, one notes that 3 is a quotient of .% (see exact sequences (l-l), (1.2)). Thus 3 is also generated by BP maps of genus one, for g a 3. Moreover, in the process of proving the exactness of sequence (1.2), Johnson shows that n,( T$) c 9 i ge~rated by Wrnqs of maximal genus; i.e., of genus g - 1. This fact will be used in determining how to obtain =. representations of ##frarn G! The case g = 2 is a bit different. In this case, Johnson showed that 9 is generated by BP maps of genus one together with BSCC maps. Note that when g = 2, any BP map is of maximal genus, thus P,( T,s) is the group of BP maps. The subgroup of .% (respectively 3 ) generated by all BSCC maps will be denoted by X (respectively 9). It will be shown that Xc ker @; therefore, BP maps of genus one will be of primary importance when calculating @I4 for all g 2 2. 1.2.
Some properties of the unit tangent bundle, which will be needed later, are reviemd in this paragraph. Let S have some Riemannian structure, inducing a IIO !! iju
R TWP
52
on each fiber IsP of the tangent bundle X9. Then T,S is the collection of all tangent vectors of unit length, i.e., T,S=
u
(UE =p/ Il4lp= 1).
P-
The unit tangent bundle T,S of S is defined similarly. It is well known that T,S is the trivial bundle, or T,S= Sx S’; hence, G, = pl( T&5)= q(S) x and fi,= At times it will be useful to think of H, asa subgroup of & in this
manner, In terms of exact squen
9 that above
14+&H,-4,
description gives (1.2.1)
where the kernel is generated by the fiber dass [z] E fi,. In the case of the closed of T$‘, however, is easily rds, the fiber class [z] E #ii vial bundle, S admits nonvanishing, mooth vector field on s it induces a on of T,S by the formula
the vector field and the section will be denoted by X) In what is to follow, X on S 41 mean a global smooth section of T,S A formula similar be used to define the diffeomorphism Qf of T,S induced by f~ 4 The derivativewoffinduces a difkomorphism of T,&,alsodenoted by Df, defined
by (l-2.3) Via dierentiation, then, an action of A on fil is obtained, and this action will be calculated. One anticipatory remark is in order before discussing winding numbers. Since & = %+‘, the action of Dfon a1 will be given by a (2g + 1) x (2g + 1) matrix with integer coefficients. Moreover, Df wiil always preserve the fiber class [ t’j E a,. If &, 6 are lifts of ui, bi to @, then the matrix of w with respect to the basis {a, a’,, . . ..&.d ,,..., gg} of fi, will have the vector (l,O, . . . ,0) as its first column. 1.3. In this section, the definition of winding numbers of curves on surfaces is given, as well as a review of some of their properties. An interpretation of winding numbers in terms of intersections in T,S is also given. This definition is particularly useful for the calculations required in the proof of Theorem 2.2. The crossed homomorphism e, is then defined, and some of its properties outlined. In particular, the behavior of the e, under composition in J# is calculated. This formula will suggest a way to define representations QXof A. First, Remark 2 on p. 221 of [l] is reproduced.
7&e mapping class group A
53
Let y be a regular closed curve on S which is parameterized by arc length (i.e., Ily’(N,W= 1 for all t ). Then T,SI, is a torus T*. Let X be a vector field on S and X(y) * Xl,,, then it is easy to see that X(y) is a curve on T*. Choose an orientation on T,S. Let a be the homotopy class of X( y) and @the homotopy class corresponding to a fiber with induced orientation; then, nl( T*) = (a, /3 1tujh~“‘/3’~= 1). Since y is parameterized by arc length, the tune 7 = ( y(t), y ‘( t )) lies on T*,and p = jslllQtE rrl( T*) for some integer m, Chiilingworth defines &y) = RP& Note that the sign of 111depends on a choice of orientation for T,S. Hodgson suggested an alternate description of this definition in terms of intersections in the homology of T,S. If X is a smooth global section of T&t, then X(S) is a surface in T,S, i.e., X(S) E H2( T,S, aT,S; ). By pojnc& duality, X(S) E if’, and the value of X(S) on any class [+]e fit is given by (X(S), j$ Choose an orientation for T,S so that (X(S), z)= 1, where [z] is the fiber class with indu orientation. Calculating X(S) on the generators a, j3 of q(r*) gives
wm
(1.34
P?= W(S), g)= 1,
(1.3.2)
(X(S),a)=(X(S),X(y))=O.
Hence (X(S), jQ = (X(S), /3*a) = m, and the following de&titian can be
o,(Y)=wm 3% with the one given This definition, with careful choices of orientations, coin by Chillingworth. This definition of wx(y) is only for curves on the sutiace S;, however, ChiWgworth shows that the notion of windin umber can be extended to the case of the closed surface if values are taken in 2g_t rather than I!. This fact will be useful in determining how the representations ex of JNcan induce representations of 2. Two properties of winding numbers are used in defining the crossed homomorphisms e, : A I--)H’. First, given two vector fields X2 ) X2 on cSi z&e is a well-defined element elJE 1y’ given by (1.3.3)
e,Jrl = o,,(Y) -w,,(Y)
for any [y] E HI. This is well defined on H, c fiI since e1,2 vanishes on the fiber class. Secondly, note that oDJ.,(fy) = w,(y). Definition 1.3.1. Let X be a vector field on S. Then e,
: A k-, H’
is defined by
e,(f)Crl=o,(fy)-o,(y) for any fdt,
ye&.
Here, and for the remainder of the paper? fv denotes &.r where _& L :ke from i automorphism of W,(S) induced by f: The fact that e,(f) E ’ fbfh6
54
R TmPP
that I, = ~~-a~(y) and the previous discussion. The most important property of the e, is the composition law:
~AfWCrl=o,WwhfhW =w,(fhr)-w,(hy)+o,(hy)-w,(y) (1.3.4)
= ex(fMrl+ ex(wCYl~ Or, more simply stated: e&tWlrl = 4f )Ihyl+ e,(JOCy19 forallf, he
(1.3.4)
and all [y] E H,. Now consider elements u E H’ as vectors in
u = Wol),
l
. . s u(q&
u(b),
l
l
- 9 W&J1
bs} is a synpkctic basis for HR.Consider the action of 36con H’ in terms UE H’ and the sympkctic representation p. The reader can convince f that the following formula is true: WY1 = [U(Ql)r-
l
l
9 u(b,)l
l
(P(h) ’ CYI)
= (bb,), . . 3 N&)1 P(W) [Yl l
l
l
(1.3.5)
= (u PW . ZYI, l
where all products are usual matrix multiplication and [ y] E HI is considered to be r, Now use (13.5) to rewrite (13.4) as c,(fTc) = df 1 P(h)+s,m
(1.3.6)
l
This formula indicates how to use the crossed homomorphism e, to define a representation @&of Before doing so, however, some remarks about exls are in order. An immediate consequence of (1.3.6) is that the exls are homomorphisms since p(h) is the identity matrix for h E .%.The following property of the homomorphisms eXIJwas proven by Johnson in [4]. His argument is repeated here.
lcralna 1.3.2. 7%ehomotmophism e,l$ is independent of the choice of vectorJeld X. Roof. Let X1, X2 be vector fields on S, and f~ .%.Then
lex,(f )-e,,(f MY1 = o,,(fY)
-o,,(Y)
- bx*(fY)
- @xz(Y)l
=~~,(fY)-~~*(fY)-[o,,(Y)-o,,(Y)l = e*z[frl - e,*2tYl = e,*r Yl - e*,z[rl = 0. The last equality holds since f e 9.
(1.3.7)
0
One reason to consider e& is that ker ax c A It will turn out that changing the choice of vector field X gives rise to conjugate representations. Hence ker Gx is independent of the choice of vector field X. Lemma 1.3.2 is just another way of stating this fact.
‘Chemapping class group 4
55
2. The rqmsentation Qi:A t, KS In this section the representationex is defined using the crossed homomorphism e,, and the dependence of ex on the vector field X is studied. The definition of winding numbers is used to show that ex calculates the action of A on fi,. As a result of this geometric interpretation,the image of A under #x is denoted by qx. The representation @I9 is then studied. In particular,the image of .%under QTis characterizedalong with ker a. Debition 2.1. Given a vector field X, the representation & :A c+ vx is given by
(2.1) where p(f) is the image offed considered to be a vector.
under the symplectic representationand e,(f) is
To see that & is indeed a representation,formula (13.6) is used in the fdlw calculation.
W) The following theorem gives the desire&geometric interpretationof the representation @,..
hf. The calculation shall be performed for a”#,all other cases are completely analogous. Suppose p( f )[a,] = (zf=, ttQ ai + ni br) in H%and consider Qf(ir’& l
l
since Qf(z) = z and Df( 6) = f< for any regular closed curve on 2%It is necessary, then, to computefz in terms of the basis {z, l&,di} for &. Note that, sincefar and (t: FzInti Ei + ni 6 ) project to the same homology class in Hz, l
l
i=l
R TroPP
56
for some integer n. To calculate n, intersect both sides of equation (2.4) with X(S) E fi’. By the definition of winding numbers, the equality
(
X(S), fx-
f mi i=l
l
i& +
ni 6
=(X(S),
l
nz)
>
becomes
(2.5) Combining (2.4), (2.3, and using the definition of {&, &}, gives
With
this
formula (2.3) becomes
cahhtio~
(2.7)
= f mi4i+ni*&+ex(f)[a,]*Z=@x(f)*[ZJ. i=l
In a similar manner it is shown that Df( 9) = @J f)
l
[f]
for all [ +] E fi, . q
An immediate consequence of Theorem 2.2 is the following
C~rdlUy 23. If Xl v X2 ate vectorfields on S, then the representations dQ,, tBs2 u”,‘c” conj44gate. Roof. By the theorem, both ax, and @x2describe the action of A on & with respect to the appropriate bases; therefore, they are conjugate. El Since the representations eX are all conjugate, they are considered a single representation a. Note, however, that the group S& depends on the choice of X. From Definition 2.1 it is easy to see that the representation @ extends the symplectic representation as promised. In order to describe ker @, it is necessary to consider @I$. For f c JJ, definition (2.1) gives
l3e mapping class group d
Here Id is the 2g x2g identity matrix. Recall that the homomorphism e,k is independent of the choice of vector field; hence, the following results and calculations are independent of the choice of vector field. The first task is to calculatethe matrix 4&(T,T$), where (a~,ai) is the BP of genus one pictured in fig. 2. If fo = T,T$ then fo(q) is conjugate to CQfor i + 2; likewise,fd & ) is conjugateto & for all i Since winding numbers are welt defined on conjugacy CI , it su&es to consider ex(fol[az] in the calculation of e,(&). The formula for calculation of winding numbers given in [I] yields ex(fo)[a,l = -2. Thus, consideringe,&) as a vector gives l%is calculation czn be used to calculate e,( T,T,-‘)
Since such maps generate .%for ga3, the desired ker @ will follow. Propebn2.4.
Let(y,~)benBPofgenwo~
withy
its kjt. Then
e,( T,T,L’) = mG a,),
l
l
l
9
(‘yra,), h b,),
l
l
l
t (Y,b,)).
Proof. For (y, 6) = (ao,a&), equation (2.9) proves the proposition. Now let (y, S) be any BP of genus one, and let f E At be such that fa= 7,fu&= SE. Then T,Ti* = f&,T;;f-‘,cuad
-(; ;g)(;
@xWJi*)-
ex;-fy; ;g,y>
= 1 e,(f-')+e,(f).df-*)+ex(h)=df-*). Id > 0
(2.10)
But O=e,(ff-')=e,(f).p(f-')+e,(f-*), so (2.10)becomes
Once it is shown that e,( fJ p( f -I)= 2(('y, a,), . . . , (y, b,)), the proof will be cornplete. The calculation preceding the proposition showed that l
e,(fo) = W-J,, 4,
- . . 9(a,, b,)) = !e,MJM,
. . 3eLhd?+J~; l
R Trapp
58
By the choke off, however, fao = -y and (2.11) becomes e,(fo)
l
df-‘1 =2((3r,
QI), . . . 9 h
(2.12)
b,))
as desired. position 2~4facilitates the calculation of e&f) for anyf= T%T&* where (ri, 86) 8te BPS of genus one.
l
Iff = Ta T&’ e,(f) = ex(TI Tc’
l
l
l
TvmTzE
whew each (yip &) is a BP of genus one,
l
l
TvmTz)
(2.13) Prooc, This is a direct consequence of Proposition 2.4 and the fact that exls is a homomorphism. c1 Corollary 2.5 can now be used to characterize ker a. 2.6. The Chillingworth subgroup %‘c § is defined by = f=T,&‘.-
I
Tyn T&l1each ( ‘yiS6i) a BP of genus one, and($,2$=OiuHJ.
Define the subgroups %“nsimilarly, the requirement being that (Cr=, 2x ) = 0 in A)rather than H, . The subgroups %g,nwill be useful in studying the structure of the groups Gg,nof [7,8]. In order to characterize ker @, Corollary 2.5 will be used together with the following fact about the group Xc 9 generated by Dehn twists on BSCCs. Chillingworth shows in [2] that, if y is a SCC, the following formula is satisfied: .
o,(Tys)=ox(B)+(Y, rB) l
o,(Y),
(2.14)
for any curve /3 on S. if y is a BSCC, then (y, ~3)= 0, and formula (2.14) showsthat TyE ker @. Since STis generated by such maps, Xc ker @.
7&emappingclassgroup At
59
Corollary 2.7. For g 2 3, % = ker #. Proof. Recalling that 9 is generated by BP maps of genus one, this is a direct consequence of Corollary2.5 and Definition 2.6. 0
Remark. The case g = 2 is special. For the purposes of this paper, it sufficesto say that it can be shown that X= ker @. Corollary 2.5 can also be used to characterizeim @I$. n2.8. Forga2, if v E (2z)2=.
Pro
thematrix (’O Id) ff is in the image of 4 under @ ifand ody
Proof. Corollary2.5 implies necessity for the subgroup of S gen of genus one. For is implies necessityfor §* Forg = 2, Corolhuy2.S, with the remarks Corollary2.7, imply necessity.Converse1 choices of BP maps of genus one, it is easy to see that contained in im(e,lJ. Since exls is a homomo
In orderto see how @ can induce repesentations of under @ must be studied. Recallthat We genus; hence it is necessary to caiculate @ on Proposition2.9. Let
T,G’
be
a BP map of genus g’. zhen
ex( T,T,‘) = 2g’((y, a,),
l
- 9(Y,a,>, l
where y is oriented so that SYsslies on its lejk Proof. The idea is to rewrite T,Ti’ in terms of BP r the calculation. Let E@, . . . , &8*-t be orientations.Then T’m-1 I+ 8 =(T,T;;)(T,,T;:) * 9 (I&T;;‘), l
of genus one, and use rves pictmed in Fig 3
l
and, using Corollary 2.5, (2.15)
Fig. 3.
R T~PP
60
However, y is homologous to &i for each i and [ y +C:LI’ Ei] = g’ [ y]. Equation (2.15) becomes l
e,( T,T,-9 = 2g’((y, 4, proving the proposition.
l
l
. 9 (Y, &)I,
(2.16)
0
Note that if (x6) is a BP of maximal genus, then g’= g - 1. Proposition2.9, with the fact that q( T,s) is generatedby BPmaps of maximal genus, imply t Lug 210. For g 3 & the matrix (i &) is in the image of w,( T$) under # ifand 0 E ((2g-2)2)‘9 1 that (p, denote @ with matrix entries taken in 2”. Then Corollary 2.10 the !I&induce representations6” (T$)cker& forn12g-2.
were discovered by Sipe, and are -2. The representations the topic of the next section. is the mapping class group of a once-punctured Onefinal remarkis in order.If sutfaoe, then there is the exact sequence
*+l,
(2.17)
where the kernel is the infinite cyclic subgroup generated by a Dehn twist about of SI Since the representation@ is trivial on BSCC maps, it follows the that oliinduces a representationof “II,,also denoted by @.All the results of this sectioncarryoverto the case of the mappingclass groupof a once-puncturedsurface.
The context in which Sipe discovered the representations& is described in this section, and her characterization of im &,, is mentioned (for details see [S]). This characterization extends to one of ST%, with the appropriate choice of vector field X0 (recall that & depends on the choice of vector field). Yhe results of Section 2 are then applied to gain further insight into the structure of the groups Can. Sipe discovered the representations 6,, while calculating the action of 2 on nth roots of the canonical bundle of the surface g Some of the results found in [7,8], are reviewed here. First, by a Chem class argument, the canonical bundle admits nth roots if and only if n ]2g - 2; therefore, it will be assumed that n I2g - 2, where g is the genus of & Sipe shows that there is a one-to-one correspondence between nth roots of the canonical bundle and the set
This topological description of nth roots is used in what follows. In order to calculate the action of & on nth roots, then, it suffices to know the action of 2 on H,( T,s; Z,). Sipe was able to calculate this action, obtaining the representations &. Recall that
The mapping class group yll
61
@,,denotes the representation @ of A with matrix entries taken in Zn. Furthermore, for n 128-2, the representation @” induces a representation of 2. The induced representation coincides with 8~~since both give the action of J@on ff,( I’,& 2,). Thus Sipe’s results give information about @,,,and it is natural to ask which results carry over to the representation @. Sipe’s characterization of im &, for example, immediately carries over to the integer-valued case. With respect to the basis {z, 4 + z, 6 + t} for fir, the following characterization of the image of 6” was given: w
3.1 (Sipe). 7%e matrix (i i) is contained in the image of & ifad
only if
Diag(B’NB) - ttE (2Z,)‘q
where BE Sp(2g; 2,) and N = (z k’). Here Diag(A) is the 2g vector whose ith component is Uiiand Bt is the transpose of B. Let X0 be the V-O~ field such that Wa(ai)=-t, @a(&)=-I for aEti Tfren, by Theorem 2.2, #% caicuiates the action of on fir with (5 &+S &+Z}* Comolluy 3.2. ne matrix (i i) is in S& ifand only if Diag(B’NB)-1t~(2b)~5 Proof. This is an immediate consequence of Theorem 2-2 and a direct generalization of Sipe’s arguments in [g] to the integer-vaiued case. Cl Note that Proposition 2.8 is the equivalent of Lemma 1 in [S], and that the results OfSection 2 reveal more of the geometric structure of @IsLA&bough Sipe’s arguments regarding im & immediately generalize to the integer-valued case, her discussion of ker @,,= Gsn does not, Let I+&@ c=Sp(2g,P) be the congruence subgroup of level n. Sipe shows that nt&an af $. ‘Ihis result does not p’(G,J = IV&“,where p’ is the symplectic rep apply to the integer-valued case since ker @c A In fact, the zesults of Section 2 can be used to further describe the structure of G”,. The fact that @(G&J = &V&m characterizes the symplectic part of C,,, while the results in Section 2 can be used to characterize G%. n x Recall the exact sequence (1.1) and Definition 2.6 of the groups @&,n. Defieritioa3.3. Let eg,” be the image of %& in .#, and iet @ be the image of the Chillingworth subgroup in J#. An immediate consequence of the definitions is the following Proposition 3.4. Gg.nfl
3 = egvn; hence the following
sequence is exact:
62
R TmPP
Roof. Definition 3.3, Corollary 2.5, and the fact that @” induces 6”.
Cl
The characterization of G”,”fl$ is virtually a direct consequence of properties of @1.,discussed in Section 2. With a little work, however, even more of the structure of Gsn is revealedNote that WM1_-2c G,, foi all R12g-2. In other words, &g-z is the subgroup
of 3 which acts trivially on nth roots of the canonical bundle for all n l2g -2. It wiil be shown tha for n even, the group Gg,,/@3,_2 is a semidirectproduct.The er of the discussion, the% will be restrictedto the case where n is even and rem be to obtain a semidirect product struare on quotients rr12g-2. The strategy n 2, and then show that these using results from of certain subgroups in enotes ker am,then pr(%!&,)= quotientsare isomorphicto G&J @&+ If !Q G,, where pr is as in sequence (1.1). Moreover,as a result of squences (l.l), (1.2), , it follows that p( %!%,) = &,“. Definition2.6 and Spe’s and C&o11 II179, giving rise to the exact squence
for all q and %c 9, the squence (3.2) modulo Ggbecomes
lthat
= ker @; therefore,the groups W&,/W,S&J % can be thought of as
subgroups of the matrix group rp%. Explicit use of Corollary 3.2, the choice of and the fact that n is even, gives the followingdescriptionsof the
(3.4)
qgJ~=@JJ~&“)=
vale,
BEN&, .
(3.3)
Note that since n is even, B E Ng,, implies B = (Id) mod 2, and Diag(B’NB) = Diag((Id)‘N(Id)) = (0,. . . , 0) mod 2.
(3.6)
Hence,if (i L) satisfies conditions (3.9, then it satisfies the conditions of Corollary 3.2. This remark also implies that the matrix (i i) is in @,J $$,) for n even and all B E Ng,*. Thus any matrix (Io B“) E @%( ?$,) can be written Uniquely as a product (i i) (i G), where both (’o B “) , (’o Id“) are in @-+(9&J. In other words, there is a semidirectproduct structure on gg,,/ Ce.Under the obvious identification of %$,/ %’ with (nh)2g the previous discussion gives the isomorophism l
(3.7)
75e mappingclassgroup A
63
Now that the desired semidirect product structure has been obtained on quotients of subgroups of A, it remains to see how this structure projects to subgroups in Recall that the representation @ induces representations of a when matrix entries are taken module 2g -2. Moreover, %sse_2/ %?c %+,,I% for n 128-2. Using the identification of %&.,/% with (nZ)*$ it is clear that
where k = (2g -2)/n. Now consider sequence (3.3) modulo the subgoup Wti~_#4E, obtaining (3.9) which becomes, after the appropriate identifications, 1+
(Zk)*&-+q&J
rg,,_,+NIkn+
(3.10)
1.
Consider %!,/ % to be the semidirect product (nZ)” >Q&+ and note that mo&g out by ((2g -2)2)** only affects the (nZ)*” factor- Hence 4p&n/ ,_2isa product as well. More explicitly,
Thissemidirectp~ductstructureonsubgroupsin~ product structure on subgroups in Propa&ion 3.5. For n emn and n 12g-2,
actually
toasemim
G,,/ @&--z s Cd&)*’>QN&,
(2s.2)/n. Woof. By equation (3.11), all that needs to be shown is that
Since ker(pr) = w,( T,& c %tig-2, the proposition is a consequence of the definitions of g&n and @gG8-2.Explicitly,
Acknowledgement This paper constitutes part of a Ph.D. thesis suf: K&led to Columbia University. The author wishes to thank his advisor Joan Birman for suggesting the problem of specializing Squier’s work, and for the ensuing encouragement and support He is also indebted to Craig Hodgson for the interpretation of winding numbers in %wns of intersection theory.
R. Trtapp
64
Refemms D.R.J.Chillingworth, Winding numbers on surfaces I, Math. Ann. 196 (1972) 218-249. D&J. Chillingworth, Winding numbers on surfaces II, Math. Ann. 199 (1972) 131-153. D. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Sot. 75 (1979) 119-125. n, An Abelian quotient of the mapping class group J$, Math. Ann. 249 (1970) 225-242. n, The structure of the Torelli group I: A finite set of generators for .& Ann. of Math. an manifolds and characteristic classes of surface bundles II, Proc. of the canonical bundle of the universal Teichmiiller cuwe and certain subgroups of a surface, Rot. Amer. Math. Sot. 97
up of an orientable surface, Israel J.